| L(s) = 1 | + 3.56·5-s − 0.438·7-s − 11-s + 2.12·13-s + 1.43·17-s − 3.56·19-s + 7.68·23-s + 7.68·25-s + 2.43·29-s − 4.56·31-s − 1.56·35-s − 7.12·37-s + 11.6·41-s + 43-s + 2.43·47-s − 6.80·49-s + 10.5·53-s − 3.56·55-s − 4·59-s + 10.2·61-s + 7.56·65-s + 4.56·67-s + 5.12·71-s + 8·73-s + 0.438·77-s + 11.1·79-s − 5·83-s + ⋯ |
| L(s) = 1 | + 1.59·5-s − 0.165·7-s − 0.301·11-s + 0.588·13-s + 0.348·17-s − 0.817·19-s + 1.60·23-s + 1.53·25-s + 0.452·29-s − 0.819·31-s − 0.263·35-s − 1.17·37-s + 1.82·41-s + 0.152·43-s + 0.355·47-s − 0.972·49-s + 1.45·53-s − 0.480·55-s − 0.520·59-s + 1.31·61-s + 0.937·65-s + 0.557·67-s + 0.608·71-s + 0.936·73-s + 0.0499·77-s + 1.25·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.626092763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626092763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 2.43T + 29T^{2} \) |
| 31 | \( 1 + 4.56T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 - 5.12T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.938450351327475631607333126964, −8.049149552609324853069437259787, −6.95774152486823281792083308522, −6.45225963908505200406576581729, −5.57194449632091368211540483714, −5.15255908504401084078037259539, −3.95874457808583122845945956781, −2.87626578287512167245090621398, −2.09061898609827686549348182995, −1.03698288676849615429422409406,
1.03698288676849615429422409406, 2.09061898609827686549348182995, 2.87626578287512167245090621398, 3.95874457808583122845945956781, 5.15255908504401084078037259539, 5.57194449632091368211540483714, 6.45225963908505200406576581729, 6.95774152486823281792083308522, 8.049149552609324853069437259787, 8.938450351327475631607333126964
